Corollary 87 ([The CM leading coefficient in known low rank]). If E/ℚ is CM and (E)≤1 , then X(E)=0 .

In rank zero gives for ^*z the scalar e_S L(E,1)/p_γ relative to ω . In rank one gives, using any free Mordell--Weil basis P , v_2( _ω(λ_ E,* ^ -1 z) (_ωP)^2 ) =v_2(e_S(L'(E,1))/(p_γH(P))) in the period-normalized sense there. Since p_γγ(a)=±Ω_0 (connected period), these are exactly the log/Haar inputs of the central positive calculations giving and, with a common multiplier of valuation v_2(γ(a)) and no smoothing. Explicitly the determinant raw valuation formula there uses the integral real boundary basis a^*(1) , subtracts ∑_ v∈S_f τ_v+lengthH^1(ℝ,T_2 E) and the dyadic -length, adds 2τ_g , and inserts v_2 of the above scalar (exponential scalar at rank zero) plus the local log-image exponent b . Haar absorbs v_2(e_S) and b , giving the Tamagawa factors and the real component correction to the whole period. Thus d(C^+(M),a^*(1)∧z)=X(E)+v_2(γ(a)) . This calculation extends scalars by flatness, and if the generic elements require denominators to lift integrally the length formula applies first to a multiple, then by homogeneity. Comparing to proves X(E)=0 . The basis P used here is a basis of the full Mordell--Weil group modulo torsion; the calculation therefore uses the regulator stipulated in Theorem 1.1.